This text is an elaborate version of a one-semester lecture course of the Budapest Semesters in Mathematics (BSM), a program for foreign students. This program was initiated and designed by Paul Erdős, László Lovász, Vera T. Sós and László Babai with the intention to offer undergraduate courses conveying the tradition of Hungarian mathematics. Altough the text discusses questions of various fields including number theory, algebra and geometry, it is centered around the real number system and the problem of measure.